E expansion in cold atoms
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e expansion in cold atoms. Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D. T. Son (INT). Ref: Phys. Rev. Lett. 97, 050403 (2006). BCS-BEC crossover and unitarity limit Formulation of e (=4-d) expansion LO & NLO results Summary and outlook.

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E expansion in cold atoms

e expansion in cold atoms

Yusuke Nishida (Univ. of Tokyo & INT)

in collaboration with D.T. Son (INT)

Ref: Phys. Rev. Lett. 97, 050403 (2006)

  • BCS-BEC crossover and unitarity limit

  • Formulation of e(=4-d) expansion

  • LO & NLO results

  • Summary and outlook

21COE WS “Strongly correlated many-body systems” 19/Jan/07


Interacting fermion systems

Interacting Fermion systems

AttractionSuperconductivity/Superfluidity

  • Metallic superconductivity (electrons)

  • Kamerlingh Onnes (1911), Tc = ~9.2 K

    • Liquid 3He

  • Lee, Osheroff, Richardson (1972), Tc = 1~2.6 mK

    • High-Tc superconductivity (electrons or holes)

  • Bednorz and Müller (1986), Tc = ~160 K

    • Atomic gases (40K, 6Li)

  • Regal, Greiner, Jin (2003), Tc ~ 50 nK

    • Nuclear matter (neutron stars):?, Tc ~ 1MeV

    • Color superconductivity (quarks):??, Tc ~ 100MeV

  • BCS theory

    (1957)


    Feshbach resonance

    Feshbach resonance

    C.A.Regal and D.S.Jin,

    Phys.Rev.Lett. 90, (2003)

    Attraction is arbitrarily tunable by magnetic field

    S-wave scattering length :[0,]

    Feshbach resonance

    a (rBohr)

    a>0

    Bound state

    formation

    molecules

    Strong coupling

    |a|

    a<0

    No bound state

    atoms

    40K

    Weak coupling

    |a|0


    Bcs bec crossover

    BCS-BEC crossover

    Eagles (1969), Leggett (1980)

    Nozières and Schmitt-Rink (1985)

    Superfluid phase

    -

    +

    0

    BCS state of atoms

    weak attraction:akF-0

    BEC of molecules

    weak repulsion:akF+0

    Strong interaction : |akF|

    • Maximal S-wave cross section Unitarity limit

    • Threshold: Ebound = 1/(ma2)  0

    Fermi gas in the strong coupling limit : akF=

    Unitary Fermi gas


    Unitary fermi gas

    Unitary Fermi gas

    George Bertsch (1999),

    “Many-Body X Challenge”

    kF-1

    r0

    V0(a)

    Atomic gas : r0=10Å << kF-1=100Å << |a|=1000Å

    What are the ground state properties of

    the many-body system composed of

    spin-1/2 fermions interacting via a zero-range,

    infinite scattering length contact interaction?

    0r0 << kF-1<< a

    kF is the only scale !

    Energy per particle

    x is independent of systems

    cf.dilute neutron matter

    |aNN|~18.5 fm >> r0~1.4 fm


    Universal parameter x

    Universal parameterx

    • Strong coupling limit

    • Perturbation akF=

    • Difficulty for theory

    • No expansion parameter

    Models

    Simulations

    Experiments

    • Mean field approx., Engelbrecht et al. (1996):x<0.59

    • Linked cluster expansion, Baker (1999):x=0.3~0.6

    • Galitskii approx., Heiselberg (2001):x=0.33

    • LOCV approx., Heiselberg (2004):x=0.46

    • Large d limit, Steel (’00)Schäfer et al. (’05):x=0.440.5

    • Carlson et al., Phys.Rev.Lett. (2003):x=0.44(1)

    • Astrakharchik et al., Phys.Rev.Lett. (2004):x=0.42(1)

    • Carlson and Reddy, Phys.Rev.Lett. (2005):x=0.42(1)

    Duke(’03): 0.74(7), ENS(’03): 0.7(1), JILA(’03): 0.5(1),

    Innsbruck(’04): 0.32(1), Duke(’05): 0.51(4), Rice(’06): 0.46(5).

    Systematic expansion forx and other

    observables (D,Tc,…) in terms ofe(=4-d)

    This talk


    2 body scattering around d 4

    2-body scattering around d=4

    iT

    iT

    2-component fermions

    local 4-Fermi interaction :

    2-body scattering at vacuum (m=0)

    (p0,p) 

    =

    n

    1

    T-matrix at d=4-e(e<<1)

    Coupling with boson

    g = (8p2e)1/2/m

    is SMALL !!!

    ig

    ig

    =

    iD(p0,p)


    Lagrangian for e expansion

    Lagrangian for e expansion

    Boson’s kinetic term is added,

    and subtracted here.

    • Hubbard-Stratonovish trans. & Nambu-Gor’kov field :

    =0 in dimensional regularization

    Ground state at finite density is superfluid :

    Expand with

    • Rewrite Lagrangian as a sum : L=L0+ L1+ L2


    Feynman rules 1

    Feynman rules 1

    • L0 :

    • Free fermion quasiparticle  and boson 

    • L1 :

    Small coupling “g” between  and 

    (g~e1/2)

    Chemical potential

    insertions (m~e)


    Feynman rules 2

    Feynman rules 2

    k

    p

    p

    =O(e)

    +

    p+k

    k

    p

    p

    p+k

    =O(em)

    +

    • L2 :

    “Counter vertices” to

    cancel 1/e singularities

    in boson self-energies

    1.

    2.

    O(e)

    O(em)


    Power counting rule of e

    Power counting rule ofe

    • Assume justified later

    • and consider to be O(1)

    • Draw Feynman diagrams using only L0 and L1

    • If there are subdiagrams of type

    • add vertices from L2 :

    • Its powers ofe will be Ng/2 + Nm

    • The only exception is= O(1) O(e)

    or

    or

    Number of m insertions

    Number of couplings “g ~e1/2”


    Thermodynamic functions at t 0

    Thermodynamic functions at T=0

    • Universal equation of state

    • Effective potential and gap equation for 0

    + O(e2)

    Veff (0,m) =

    +

    +

    O(e)

    O(1)

    • Universal numberxaround d=4

    Systematic

    expansion !


    Quasiparticle spectrum

    Quasiparticle spectrum

    p-k

    k-p

    -iS(p) =

    +

    p

    p

    p

    p

    k

    k

    • O(e) fermion self-energy

    • Fermion dispersion relation : w(p)

    Aroundminimum

    Expansion over 4-d

    Energy gap :

    Location of min. :

    0


    Extrapolation to d 3 from d 4 e

    Extrapolation to d=3 from d=4-e

    • Keep LO & NLO resultsand extrapolate to e=1

    NLO

    corrections

    are small

    5 ~ 35 %

    Good agreement with recent Monte Carlo data

    J.Carlson and S.Reddy,

    Phys.Rev.Lett.95, (2005)

    cf. extrapolations from d=2+e

    NLO are 100 %


    Matching of two expansions in x

    Matching of two expansions in x

    • Borel transformation + Padé approximants

    Expansion around 4d

    x

    ♦=0.42

    2d boundary condition

    2d

    • Interpolated results to 3d

    4d

    d


    Summary

    Summary

    • Systematic expansions over e=4-d

      • Unitary Fermi gas around d=4 becomes

      • weakly-interacting system of fermions & bosons

    • LO+NLO results onx, D, e0

      • NLO corrections around d=4 are small

      • Extrapolations to d=3 agree with recent MC data

    • Future problems

      • Large order behavior + NN…LO corrections

      • More understanding Precise determination

    Picture of weakly-interacting fermionic &

    bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3


    Back up slides

    Back up slides


    Specialty of d 4 and 2

    Specialty of d=4 and 2

    Z.Nussinov and S.Nussinov, cond-mat/0410597

    2-body wave function

    Normalization at unitarity a

    diverges at r0 for d4

    Pair wave function is concentrated near its origin

    Unitary Fermi gas for d4 is free “Bose” gas

    At d2, any attractive potential leads to bound states

    “a” corresponds to zero interaction

    Unitary Fermi gas for d2 is free Fermi gas


    Specialty of d 4 and d 2

    Specialty of d=4 and d=2

    iT

    2-component fermions

    local 4-Fermi interaction :

    2-body scattering in vacuum (m=0)

    (p0,p) 

    =

    n

    1

    T-matrix at arbitrary spatial dimension d

    “a”

    Scattering amplitude has zeros at d=2,4,…

    Non-interacting limits


    T matrix around d 4 and 2

    T-matrix around d=4 and 2

    iT

    iT

    T-matrix at d=4-e(e<<1)

    Small coupling b/w fermion-boson

    g = (8p2e)1/2/m

    ig

    ig

    =

    iD(p0,p)

    T-matrix at d=2+e(e<<1)

    Small coupling b/w fermion-fermion

    g = (2pe/m)1/2

    ig2

    =


    Unitary fermi gas at d 3

    Unitary Fermi gas at d≠3

    g

    g

    d=4

    • d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2

    Strong coupling

    Unitary regime

    BEC

    BCS

    -

    +

    • d2 : Weakly-interacting system of fermions, their coupling is g~(d-2)

    d=2

    Systematic expansions forx and other observables (D, Tc, …) in terms of “4-d” or “d-2”


    Expansion over e d 2

    Expansion over e=d-2

    Lagrangian

    Power counting rule of

    • Assume justified later

    • and consider to be O(1)

    • Draw Feynman diagrams using only L0 and L1

    • If there are subdiagrams of type

    • add vertices from L2 :

    • Its powers ofe will be Ng/2


    Nnlo correction for x

    NNLO correction for x

    Arnold, Drut, and Son, cond-mat/0608477

    • O(e7/2) correction for x

    • Borel transformation + Padé approximants

    x

    • Interpolation to 3d

    • NNLO 4d + NLO 2d

    • cf. NLO 4d + NLO 2d

    NLO 4d

    NLO 2d

    d

    NNLO 4d


    Critical temperature

    Critical temperature

    • Gap equation at finite T

    Veff = + + + minsertions

    • Critical temperature from d=4 and 2

    NLO correctionis small ~4 %

    Simulations :

    • Lee and Schäfer (’05): Tc/eF < 0.14

    • Burovski et al. (’06): Tc/eF = 0.152(7)

    • Akkineni et al. (’06): Tc/eF 0.25

    • Bulgac et al. (’05): Tc/eF = 0.23(2)


    Matching of two expansions t c

    Matching of two expansions (Tc)

    Tc/eF

    4d

    2d

    d

    • Borel + Padé approx.

    • Interpolated results to 3d


    E expansion in critical phenomena

    eexpansion in critical phenomena

    Critical exponents of O(n=1) 4 theory(e=4-d1)

    • Borel summation with conformal mapping

    • g=1.23550.0050 & =0.03600.0050

    • Boundary condition (exact value at d=2)

    • g=1.23800.0050 & =0.03650.0050

    e expansion is

    asymptotic series

    but works well !

    How about our case???


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